Generalized Intuitionistic Multiplicative Fuzzy Calculus Theory and Applications by Shan Yu & Zeshui Xu

Author:Shan Yu & Zeshui Xu
Language: eng
Format: epub
ISBN: 9789811556128
Publisher: Springer Singapore

Theorem 4.3

(Yu and Xu 2016a) If is a subtraction derivable function of , then

Proof. By Definition 2.​5 and Theorem 4.2, we have

Since

and

then

Theorem 4.3 tells us that the subtraction primitive function (or antiderivative) of exists, which can be denoted by .

Now, we use it to evaluate the subtraction definite integrals without having to calculate the limits of and call it the fundamental theorem of calculus in the context of generalized intuitionistic multiplicative fuzzy circumstances.

Theorem 4.4

(Yu and Xu 2016a; The fundamental theorem of calculus) If the subtraction definite integral of exists over any closed interval , and is any subtraction primitive function of over , then

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